\[\begin{align*} \boxed{ e^{\ln x} = x } \end{align*}\]
What is $e$?
\( e \) is a special number in math, approximately equal to 2.718. It is also called Euler’s constant and is the base of a logarithm called the natural logarithm. You can think of it as the number that makes many things in math work nicely, like growth or change. You can also think of it as just another constant, like $\pi$.
What is $\ln x$?
The natural logarithm, written as \(\ln x\), is the inverse operation of exponentiation with base \( e \). This means that \( \ln x \) asks the question, “To what power must we raise \( e \) in order to get \( x \)?” In other words,
\[\begin{align*} \ln x = y \quad \text{means} \quad e^y = x \end{align*}\]
So, for example, if \( x = 7.389 \), then \( \ln 7.389 \approx 2 \), because \( e^2 \approx 7.389 \).
What does $e^{\ln x}$ mean?
When you see the expression \( e^{\ln x} \), it means you are raising \( e \) to the power of \( \ln x \). But here’s the catch: since the natural logarithm and \( e \) are connected in a special way, raising \( e \) to the power of \( \ln x \) just asks, “When you raise $e$ to the power that you would need to raise $e$ to in order to get $x$, what do you get?” This sounds like a mouthful, but it, essentially, boils down to
\[\begin{align*} e^{\ln x} = x \end{align*}\]
